【作者】 李林；

【导师】 葛斌；

【作者基本信息】 哈尔滨工程大学 ， 数学， 2021， 硕士

【摘要】 群集行为是指大量粒子通过相互作用所形成的有序集群运动,是生物、控制、数学等领域研究者对生物群体行为的具体描述。近年来,群集行为的研究引起了国内外众多学者的关注,其中多数群集系统采用渐近收敛方式,收敛时间趋于无穷。因此,粒子系统的收敛快慢成为了衡量系统性能的一个重要指标。Cucker-Smale系统通过鉴借鸟群、鱼群等达到群集的方式,构建了自推进粒子系统,即选取增加控制项的方法使粒子系统的群集运动得以优化。然而,粒子在有限时间内进行信息传递的过程中,不可避免会受到外界的干扰。为了研究真实系统的稳定性,本文首先介绍了群集的基本概念、常微分方程等相关知识,然后引入了本文的主要研究内容。具体如下:首先,针对带有领导者的确定Cucker-Smale模型在有限时间内的群集行为,建立了带有两个非线性控制项的Cucker-Smale数学机制。基于有限时间稳定性理论,利用矩阵理论、代数图论和Lyapunov函数等方法,证明了定理中粒子发生群集的充分条件的合理性。通过数值实验,验证了所得结果的正确性。其次,针对带有领导者的随机Cucker-Smale模型在有限时间内的群集行为,采用了在确定模型中加入噪声项和非线性控制项的方法,克服了需要较长时间粒子才能群集的困难。利用随机微分方程的有限时间稳定性理论、比较定理、Lyapunov函数方法和代数图论等,证明了所给定理中粒子发生群集的充分条件的合理性。最后对粒子系统进行了数值仿真,验证了理论结果的正确性和可靠性。更多还原

【Abstract】 Flocking behavior refers to the orderly flock motion formed by a large number of particles through interaction.This behavior is a specific description of the behavior of biological groups by researchers in the fields of biology,control,and mathematics.Recently,the study of flock behavior has attracted the attention of many scholars at home and abroad.Most of the flock systems adopt asymptotic convergence,and the convergence time tends to infinity.Therefore,the convergence speed of the particle system has become an important indicator to measure the performance of the system.The Cucker-Smale system constructs a self-propelled particle system by identifying birds and fish schools to achieve flocking.That is to say,the method of adding control items to optimize the flock motion of the particle system.However,particles will inevitably be disturbed by the outside world in the process of information transmission within a limited time.In order to study the stability of the real system,this thesis first introduces the basic concepts of flocks,ordinary differential equations and other related knowledge,and then introduces the main research content of this thesis.Details as follows:Firstly,to determine the flocking behavior of the Cucker-Smale model with a leader in a limited time,a Cucker-Smale mathematical mechanism with two nonlinear control terms is established.Based on the finite-time stability theory,matrix theory,algebraic graph theory and Lyapunov function are used to prove the rationality of the sufficient conditions for the flocking of particles in the theorem.Numerical experiments verify the correctness of the obtained results.Secondly,aiming at the flocking behavior of the random Cucker-Smale model with a leader in a finite time,the method of adding noise terms and nonlinear control terms to the deterministic model is adopted,which overcomes the difficulty of requiring a long time for particles to flock.Using the finite-time stability theory of stochastic differential equations,comparison theorem,Lyapunov function method and algebraic graph theory,etc.,the rationality of the sufficient conditions for the flocking of particles in the given theorem is proved.Finally,a numerical simulation of the particle system is carried out to verify the correctness and reliability of the theoretical results.更多还原